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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Neumann Boundary Condition.</dfn> We now consider a set of new BCs: (insulated)</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u_x(0,t)=0,\quad u_x(L,t)=0.
\end{equation*}
</div>
<p class="continuation">This means that both ends are insulated, and no heat flows through. Following the same setting, we get the eigenvalue problem for <span class="process-math">\(X(x)\)</span> as</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X''+\lambda X=0,\quad X'(0)=X'(L)=0,
\end{equation*}
</div>
<p class="continuation">From Example 2 in Section <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "nbcex" missing or not unique]</code>, we have only nonnegative eigenvalues <span class="process-math">\(\lambda_n\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\lambda_n={\left(\frac{n\pi}{L}\right)^2},\quad {X_n(x)}={\cos\frac{n\pi x}{L}},\quad n=\textcolor{red}{0},1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">The solution for <span class="process-math">\(T(t)\)</span> remains the same</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
T_n(t)=C_n\cdot\exp\left[-\left(\frac{n\pi\alpha}{L}\right)^2t\right], \quad n=\textcolor{red}{0},1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">which leads to the formal solution</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=C_0 + \sum_{n=1}^{\infty}X_n(x)T_n(t)=C_0 + \sum_{n=1}^{\infty}C_n e^{-\frac{n^2\pi^2\alpha^2}{L^2}t} \cos \frac{n\pi x}{L},
\end{equation*}
</div>
<p class="continuation">Finally, by fitting in the initial condition, <span class="process-math">\(C_n\)</span> can be determined as the Fourier cosine coefficient for the even half-range expansion of <span class="process-math">\(f(x)\text{,}\)</span> i.e.,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_0=\frac{1}{2}\frac{2}{L}\int_{0}^{L} f(x)~\textrm{d}x=\frac{1}{L}\int_{0}^{L} f(x)~\textrm{d}x,\qquad C_n=\frac{2}{L}\int_{0}^{L} f(x)\cos \frac{n\pi x}{L}~\textrm{d}x, \quad n=1,2,3,\cdots.
\end{equation*}
</div>
<span class="incontext"><a href="sec7_8.html#p-416" class="internal">in-context</a></span>
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